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XLII. On the Calculation of the Numerical Values of a certain 
class of Multiple and Definite Integrals. By Sir Wit114m 
Rowan Hamitton, LL.D., M.RLA., F.RAS. &e., An- 
drews’ Professor of Astronomy in the University of Dublin, 
and Royal Astronomer of Ireland*. 
Srcrion I. 
LE.| A det results, in part numerical, of which a sketch is 
here to be given, may serve to illustrate some points 
in the theory of functions of large numbers, and in that of defi- 
nite and multiple integrals. In stating them, it will be conve- 
nient to employ a notation which I have formerly published, and 
have often found to be useful; namely the following, 
it 
l= (aes. _ pila eee 
0 
é 
Igt={ FY ae ee Brera 00 
: 
with which I am now disposed to combine this other symbol, 
2) 
s=( dt 3 . . . ° . . ° (2) 
z 
in such a manner as to write, 
Tofee ! an’, eg rh ey 
or more fully, 
and therefore 
fe 2] 
I; + Je = dt. . . . . . ° (3) 
0 
I shall also retain, for the present, the known notation of Van- 
dermonde for factorials, which has been described and used by 
Lacroix, and in which, for any positive whole value of n, 
[a]"=2(4—1)(e@—2)...(w—n+1); . - - - 4 
so that there are the transformations, 
(a]"=[#]"[e2—m]"-"= [2]"*": [w—n]”, &e. 5 (4)! 
which are extended by definition to the case of null and negative 
indices, and give, in particular, oe 
cay" 
For example, 
(hta\icsace. [np [Glare sie © sis dy (D) 
* Communicated by the Author. 
